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Theorem ablgrp 13098
Description: An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
Assertion
Ref Expression
ablgrp  |-  ( G  e.  Abel  ->  G  e. 
Grp )

Proof of Theorem ablgrp
StepHypRef Expression
1 isabl 13097 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
21simplbi 274 1  |-  ( G  e.  Abel  ->  G  e. 
Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148   Grpcgrp 12882  CMndccmn 13093   Abelcabl 13094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-abl 13096
This theorem is referenced by:  ablgrpd  13099  ablinvadd  13118  ablsub2inv  13119  ablsubadd  13120  ablsub4  13121  abladdsub4  13122  abladdsub  13123  ablpncan2  13124  ablpncan3  13125  ablsubsub  13126  ablsubsub4  13127  ablpnpcan  13128  ablnncan  13129  ablnnncan  13131  ablnnncan1  13132  ablsubsub23  13133
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